TMDD Part 1: Understanding Target-Mediated Drug Disposition

Authors

Vijay Ivaturi

Srayant Gayam

1 Introduction

This is the first part of a comprehensive tutorial series on Target-Mediated Drug Disposition (TMDD) modeling in Pumas. In this part, we focus on building intuition for TMDD before diving into complex models.

Tutorial Series Overview:

Part 1: Understanding TMDD (this tutorial) - Conceptual foundations and parameter meaning 1b. Part 1b: TMDD Dynamics Through the Four Phases - Deep dive into phase characterization
Part 2: The Full TMDD Model - Complete mechanistic model implementation
Part 3: TMDD Approximations - QE, QSS, and Michaelis-Menten models
Part 4: Special Cases and Extensions - Advanced TMDD scenarios
Part 5: Practical Workflows - Fitting, diagnostics, and model selection

2 Learning Objectives

By the end of this tutorial, you will be able to:

Explain what TMDD is and why it matters in drug development
Recognize TMDD signatures in pharmacokinetic data
Understand the biological basis of ligand-receptor binding kinetics
Interpret key TMDD parameters ($K_D$, $k_{on}$, $k_{off}$, $k_{syn}$, $k_{deg}$, $k_{int}$)
Distinguish between linear and nonlinear PK behavior

3 Libraries

using Pumas
using PumasUtilities
using AlgebraOfGraphics
using CairoMakie
using DataFrames
using Random

4 What is Target-Mediated Drug Disposition?

Target-Mediated Drug Disposition (TMDD) is a pharmacokinetic phenomenon where the binding of a drug (ligand) to its pharmacological target (receptor) significantly influences the drug’s disposition in the body. This concept was formally introduced by Levy (1994).

The key insight is that for drugs with high affinity to their targets, the target itself becomes a meaningful elimination pathway. When target binding is saturable and the target concentration is pharmacologically relevant, the drug exhibits nonlinear pharmacokinetics.

4.1 When Does TMDD Matter?

TMDD is most commonly observed with:

Monoclonal antibodies (mAbs) binding to soluble or membrane-bound targets
High-affinity small molecules with potent target engagement
Therapeutic proteins interacting with cell surface receptors
Antibody-drug conjugates (ADCs)

The TMDD Signature

A hallmark of TMDD is dose-dependent clearance: at low doses, clearance is high because target-mediated elimination dominates. At high doses (when targets are saturated), clearance decreases to reflect only non-specific elimination pathways.

5 The Biology of Ligand-Receptor Binding

Before we can model TMDD, we need to understand the underlying biology. Consider a drug (the ligand, $L$) entering the body and encountering its pharmacological target (the receptor, $R$).

5.1 The Binding Process

The ligand-receptor interaction follows reversible binding kinetics:

\[ L + R \underset{k_{off}}{\stackrel{k_{on}}{\rightleftharpoons}} LR \]

Where:

$L$ = Free ligand (drug) concentration
$R$ = Free receptor (target) concentration
$LR$ = Ligand-receptor complex concentration
$k_{on}$ = Association rate constant (units: concentration$^{-1}$ time$^{-1}$)
$k_{off}$ = Dissociation rate constant (units: time$^{-1}$)

The equilibrium dissociation constant, $K_D$, characterizes the binding affinity:

\[ K_D = \frac{k_{off}}{k_{on}} \]

A lower $K_D$ indicates higher binding affinity. Typical values for mAbs are in the nanomolar to picomolar range.

5.2 Receptor Dynamics

In the body, receptors are not static. They are continuously synthesized and degraded:

\[ \emptyset \xrightarrow{k_{syn}} R \xrightarrow{k_{deg}} \emptyset \]

Where:

$k_{syn}$ = Zero-order synthesis rate of receptor (units: concentration/time)
$k_{deg}$ = First-order degradation rate constant of receptor (units: time$^{-1}$)

At baseline (before drug administration), the receptor reaches a steady-state concentration:

\[ R_0 = \frac{k_{syn}}{k_{deg}} \]

Parameter Interpretation

The baseline receptor concentration $R_0$ is determined by the ratio of synthesis to degradation rates. This value is crucial for understanding the magnitude of TMDD effects.

5.3 Complex Fate: Internalization

The ligand-receptor complex does not persist indefinitely. It is typically internalized (endocytosed) and degraded at a rate $k_{int}$:

\[ LR \xrightarrow{k_{int}} \text{degradation products} \]

The relative magnitude of $k_{int}$ versus $k_{deg}$ determines whether target engagement accelerates or decelerates elimination:

If $k_{int} > k_{deg}$: The complex is cleared faster than free receptor, leading to increased clearance at low doses
If $k_{int} < k_{deg}$: The complex is more stable, potentially leading to accumulation

6 Visualizing Linear vs. TMDD Kinetics

Let’s build intuition by comparing linear PK (no target binding) with TMDD kinetics.

6.1 A Simple Linear Model

First, we’ll create a simple one-compartment model with linear elimination:

linear_model = @model begin
    @param begin
        tvCl ∈ RealDomain(lower = 0)
        tvVc ∈ RealDomain(lower = 0)
    end

    @pre begin
        Cl = tvCl
        Vc = tvVc
    end

    @dynamics begin
        Central' = -Cl / Vc * Central
    end

    @derived begin
        Concentration = @. Central / Vc
    end
end

PumasModel
  Parameters: tvCl, tvVc
  Random effects:
  Covariates:
  Dynamical system variables: Central
  Dynamical system type: Matrix exponential
  Derived: Concentration
  Observed: Concentration

6.2 A Simple TMDD Model

Now, let’s create a one-compartment TMDD model with target-mediated elimination:

tmdd_simple_model = @model begin
    @param begin
        tvCl ∈ RealDomain(lower = 0)      # Linear clearance
        tvVc ∈ RealDomain(lower = 0)      # Central volume
        tvKsyn ∈ RealDomain(lower = 0)    # Receptor synthesis rate
        tvKdeg ∈ RealDomain(lower = 0)    # Receptor degradation rate
        tvKon ∈ RealDomain(lower = 0)     # Association rate
        tvKoff ∈ RealDomain(lower = 0)    # Dissociation rate
        tvKint ∈ RealDomain(lower = 0)    # Internalization rate
    end

    @pre begin
        Cl = tvCl
        Vc = tvVc
        Ksyn = tvKsyn
        Kdeg = tvKdeg
        Kon = tvKon
        Koff = tvKoff
        Kint = tvKint
    end

    @dosecontrol begin
        bioav = (Ligand = 1 / tvVc,)  # Dose enters as concentration
    end

    @init begin
        Receptor = Ksyn / Kdeg  # Baseline receptor level (R0)
    end

    @vars begin
        R0 := Ksyn / Kdeg
        Ltot := Ligand + Complex
        Rtot := Receptor + Complex
    end

    @dynamics begin
        Ligand' = -(Cl / Vc) * Ligand - Kon * Ligand * Receptor + Koff * Complex
        Receptor' = Ksyn - Kdeg * Receptor - Kon * Ligand * Receptor + Koff * Complex
        Complex' = Kon * Ligand * Receptor - Koff * Complex - Kint * Complex
    end

    @derived begin
        Free_Ligand = @. Ligand
        Free_Receptor = @. Receptor
        Complex_Conc = @. Complex
    end
end

PumasModel
  Parameters: tvCl, tvVc, tvKsyn, tvKdeg, tvKon, tvKoff, tvKint
  Random effects:
  Covariates:
  Dynamical system variables: Ligand, Receptor, Complex
  Dynamical system type: Nonlinear ODE
  Derived: Free_Ligand, Free_Receptor, Complex_Conc
  Observed: Free_Ligand, Free_Receptor, Complex_Conc

6.3 Comparing the Models

Let’s simulate both models across a range of doses to see how TMDD manifests as nonlinear kinetics:

Show/Hide Code

# Parameters for linear model
param_linear = (tvCl = 0.01, tvVc = 3.0)

# Parameters for TMDD model based on Gibiansky et al.
# These values represent typical mAb parameters
param_tmdd = (
    tvCl = 0.006,     # Linear clearance: ~0.15 L/day
    tvVc = 3.0,       # Central volume: 3 L (plasma volume)
    tvKsyn = 0.04,    # Synthesis rate (nM/hr) → ~1 nmol/day
    tvKdeg = 0.008,   # Degradation rate (hr⁻¹) → R0 = 5 nM, t½ ~87 hr
    tvKon = 0.33,     # Association rate (nM⁻¹ hr⁻¹) → ~8 (nM)⁻¹/day
    tvKoff = 0.33,    # Dissociation rate (hr⁻¹) → KD = 1 nM
    tvKint = 0.002,   # Internalization rate (hr⁻¹) → ~0.04/day
)

# Create populations with different doses
# Wide dose range to show saturation of target-mediated elimination
doses = [0.5, 5, 50, 500]  # nmol (spanning 3 orders of magnitude)
Random.seed!(42)

pop_linear = [
    Subject(
        id = i,
        events = DosageRegimen(dose, time = 0),
        observations = (Concentration = nothing,),
    ) for (i, dose) in enumerate(doses)
]

pop_tmdd = [
    Subject(
        id = i,
        events = DosageRegimen(dose, time = 0, cmt = 1),
        observations = (
            Free_Ligand = nothing,
            Free_Receptor = nothing,
            Complex_Conc = nothing,
        ),
    ) for (i, dose) in enumerate(doses)
]

# Simulate (extended time for slower mAb kinetics)
obstimes = 0.1:2:672  # Up to 4 weeks

sim_linear = simobs(linear_model, pop_linear, param_linear, obstimes = obstimes)
sim_tmdd = simobs(tmdd_simple_model, pop_tmdd, param_tmdd, obstimes = obstimes)

Simulated population (Vector{<:Subject})
  Simulated subjects: 4
  Simulated variables: Free_Ligand, Free_Receptor, Complex_Conc

Now let’s visualize the difference:

Show/Hide Code

# Convert simulation results to DataFrames
df_linear = DataFrame(sim_linear)
df_tmdd = DataFrame(sim_tmdd)

# Create dose label mapping
dose_labels = DataFrame(id = string.(1:4), Dose = string.(doses) .* " nmol")

# Add dose labels and calculate dose-normalized concentrations
df_linear = innerjoin(df_linear, dose_labels, on = :id)
df_linear.Conc_Norm =
    df_linear.Concentration ./ parse.(Float64, replace.(df_linear.Dose, " nmol" => ""))
df_linear.Model .= "Linear PK"

df_tmdd = innerjoin(df_tmdd, dose_labels, on = :id)
df_tmdd.Conc_Norm =
    df_tmdd.Free_Ligand ./ parse.(Float64, replace.(df_tmdd.Dose, " nmol" => ""))
df_tmdd.Model .= "TMDD"

# Combine for faceted plot
df_combined = vcat(
    select(df_linear, :time, :Dose, :Conc_Norm, :Model),
    select(df_tmdd, :time, :Dose, :Conc_Norm, :Model),
)

plt =
    data(df_combined) *
    mapping(
        :time => "Time (hr)",
        :Conc_Norm => "Dose-Normalized Concentration";
        color = :Dose,
        layout = :Model,
    ) *
    visual(Lines; linewidth = 2)

draw(plt; axis = (yscale = Makie.pseudolog10,))

Comparison of linear PK (left) versus TMDD kinetics (right). Linear PK shows superimposable dose-normalized curves. TMDD shows dose-dependent behavior with faster decline at lower doses.

Key Observation

In linear PK (left panel), dose-normalized concentrations superimpose regardless of dose level.

In TMDD (right panel), dose-normalized curves diverge: lower doses show faster terminal decline because target-mediated clearance dominates.

7 Recognizing TMDD in Your Data

When analyzing PK data, several signatures suggest TMDD may be present:

7.1 Dose-Dependent Half-Life

The apparent terminal half-life changes with dose:

At low doses: Short half-life (target-mediated elimination dominates)
At high doses: Long half-life (linear elimination dominates after target saturation)

Show/Hide Code

plt =
    data(df_tmdd) *
    mapping(
        :time => "Time (hr)",
        :Free_Ligand => "Free Ligand Concentration";
        color = :Dose => "Dose",
    ) *
    visual(Lines; linewidth = 2)

draw(
    plt;
    axis = (yscale = Makie.pseudolog10, title = "TMDD: Dose-Dependent Terminal Kinetics"),
)

Concentration-time profiles showing dose-dependent terminal slope characteristic of TMDD.

7.2 Non-Parallel Decline

On a semi-log plot, concentration-time curves at different doses are not parallel during the terminal phase. This contrasts with linear kinetics where all doses show parallel decline.

7.3 Rapid Initial Decline Followed by Slower Terminal Phase

TMDD often produces a characteristic multiphasic profile. Peletier & Gabrielsson (2012) identified four distinct phases (A, B, C, D) in TMDD concentration-time curves:

Phase A: Rapid initial binding (target association)
Phase B: Linear decline with saturated target
Phase C: Non-linear transition as target recovers
Phase D: Terminal linear phase (often internalization-limited)

This four-phase characterization is explored in detail in Part 1b: TMDD Dynamics Through the Four Phases, which provides comprehensive visualizations and parameter identification guidance.

7.4 Disproportionate Increase in AUC with Dose

For a linear drug, AUC increases proportionally with dose. With TMDD, AUC increases more than proportionally at higher doses because target-mediated clearance becomes saturated.

8 Understanding TMDD Parameters

Let’s explore how each parameter affects the PK profile through interactive simulations.

8.1 Effect of Binding Affinity ($K_D$)

The dissociation constant $K_D = k_{off}/k_{on}$ determines binding affinity. Lower $K_D$ means tighter binding.

Show/Hide Code

# Single dose, varying KD
single_pop = [
    Subject(
        id = 1,
        events = DosageRegimen(50, time = 0, cmt = 1),
        observations = (
            Free_Ligand = nothing,
            Free_Receptor = nothing,
            Complex_Conc = nothing,
        ),
    ),
]

kd_values = [0.1, 1.0, 5.0, 20.0]  # nM; Lower = higher affinity

# Build DataFrame with all KD variations
df_kd = DataFrame()
for kd in kd_values
    param_kd = (
        tvCl = 0.006,
        tvVc = 3.0,
        tvKsyn = 0.04,
        tvKdeg = 0.008,
        tvKon = 0.33,
        tvKoff = kd * 0.33,  # Koff = KD * Kon
        tvKint = 0.002,
    )
    sim_kd = simobs(tmdd_simple_model, single_pop, param_kd, obstimes = obstimes)
    sim_df = DataFrame(sim_kd)
    sim_df.KD .= "KD = $kd nM"
    append!(df_kd, sim_df)
end

plt =
    data(df_kd) *
    mapping(
        :time => "Time (hr)",
        :Free_Ligand => "Free Ligand Concentration";
        color = :KD => "KD (nM)",
    ) *
    visual(Lines; linewidth = 2)

draw(plt; axis = (yscale = Makie.pseudolog10, title = "Effect of Binding Affinity (KD)"))

Effect of binding affinity (KD) on TMDD profiles. Tighter binding (lower KD) leads to more pronounced target-mediated elimination.

8.2 Effect of Receptor Concentration ($R_0$)

The baseline receptor level $R_0 = k_{syn}/k_{deg}$ determines the magnitude of TMDD. Higher receptor concentrations mean more drug can be eliminated via target binding.

Show/Hide Code

r0_values = [1, 5, 15, 30]  # Different R0 levels (nM)

# Build DataFrame with all R0 variations
df_r0 = DataFrame()
for r0 in r0_values
    param_r0 = (
        tvCl = 0.006,
        tvVc = 3.0,
        tvKsyn = r0 * 0.008,  # Ksyn = R0 * Kdeg
        tvKdeg = 0.008,
        tvKon = 0.33,
        tvKoff = 0.33,
        tvKint = 0.002,
    )
    sim_r0 = simobs(tmdd_simple_model, single_pop, param_r0, obstimes = obstimes)
    sim_df = DataFrame(sim_r0)
    sim_df.R0 .= "R0 = $r0 nM"
    append!(df_r0, sim_df)
end

plt =
    data(df_r0) *
    mapping(
        :time => "Time (hr)",
        :Free_Ligand => "Free Ligand Concentration";
        color = :R0 => "R0 (nM)",
    ) *
    visual(Lines; linewidth = 2)

draw(
    plt;
    axis = (yscale = Makie.pseudolog10, title = "Effect of Baseline Receptor Level (R0)"),
)

Effect of baseline receptor concentration (R0) on TMDD profiles. Higher R0 leads to greater target-mediated elimination.

8.3 Effect of Internalization Rate ($k_{int}$)

The internalization rate determines how quickly the ligand-receptor complex is eliminated.

Show/Hide Code

kint_values = [0.001, 0.002, 0.008, 0.02]  # hr⁻¹

# Build DataFrame with all kint variations
df_kint = DataFrame()
for kint in kint_values
    param_kint = (
        tvCl = 0.006,
        tvVc = 3.0,
        tvKsyn = 0.04,
        tvKdeg = 0.008,
        tvKon = 0.33,
        tvKoff = 0.33,
        tvKint = kint,
    )
    sim_kint = simobs(tmdd_simple_model, single_pop, param_kint, obstimes = obstimes)
    sim_df = DataFrame(sim_kint)
    sim_df.kint .= "kint = $kint hr⁻¹"
    append!(df_kint, sim_df)
end

plt =
    data(df_kint) *
    mapping(
        :time => "Time (hr)",
        :Free_Ligand => "Free Ligand Concentration";
        color = :kint => "kint (hr⁻¹)",
    ) *
    visual(Lines; linewidth = 2)

draw(
    plt;
    axis = (yscale = Makie.pseudolog10, title = "Effect of Internalization Rate (kint)"),
)

Effect of internalization rate on TMDD profiles. Faster internalization increases target-mediated clearance.

9 Parameter Summary Table

The following table summarizes the key TMDD parameters and their biological interpretation:

Parameter	Symbol	Units	Biological Meaning
Association rate	$k_{on}$	nM$^{-1}$hr$^{-1}$	Rate of ligand-receptor binding
Dissociation rate	$k_{off}$	hr$^{-1}$	Rate of complex dissociation
Dissociation constant	$K_D$	nM	Binding affinity ($K_D = k_{off}/k_{on}$)
Synthesis rate	$k_{syn}$	nM/hr	Rate of receptor production
Degradation rate	$k_{deg}$	hr$^{-1}$	Rate of free receptor turnover
Internalization rate	$k_{int}$	hr$^{-1}$	Rate of complex elimination
Baseline receptor	$R_0$	nM	Steady-state receptor level ($R_0 = k_{syn}/k_{deg}$)

Typical Parameter Ranges for Monoclonal Antibodies

Based on Gibiansky et al. (2008) and Gibiansky & Gibiansky (2009), along with clinical experience with therapeutic mAbs:

Parameter	Typical Range	Units	Notes
$CL$	0.1 - 0.5	L/day	Non-specific linear clearance
$V_c$	2 - 5	L	Approximately plasma volume
$K_D$	0.01 - 10	nM	Very high affinity
$K_{SS}$	0.1 - 10	nM	$K_{SS} = (k_{off} + k_{int})/k_{on}$
$k_{on}$	0.1 - 10	nM$^{-1}$day$^{-1}$	Association rate
$k_{off}$	0.01 - 10	day$^{-1}$	Dissociation rate
$R_0$	0.1 - 100	nM	Target-dependent
$k_{deg}$	0.1 - 1	day$^{-1}$	Receptor turnover
$k_{int}$	0.01 - 1	day$^{-1}$	Often similar to $k_{deg}$

The ratio of $k_{int}/k_{deg}$ determines whether binding increases or decreases target turnover.

10 When to Suspect TMDD

Consider TMDD modeling when you observe:

Dose-dependent clearance decreasing with increasing dose
Non-superimposable dose-normalized profiles
Rapid initial decline at low doses followed by slower terminal phase
Greater than dose-proportional AUC increases
Biological plausibility: high-affinity target with pharmacologically relevant concentrations

Not All Nonlinearity is TMDD

Other sources of nonlinear PK include:

Saturable absorption (flip-flop kinetics)
Saturable protein binding
Capacity-limited hepatic metabolism
Time-dependent autoinduction or inhibition
Formation of anti-drug antibodies

Careful mechanistic analysis is needed to distinguish these phenomena.

11 Summary

In this tutorial, we covered the conceptual foundations of TMDD:

TMDD occurs when high-affinity target binding significantly affects drug disposition
Key components are the ligand, receptor, and complex, each with associated rate constants
The signature of TMDD is dose-dependent clearance and non-superimposable profiles
Parameters have biological meaning: $K_D$ reflects affinity, $R_0$ reflects target abundance, $k_{int}$ reflects complex elimination

In Part 2, we will implement the full mechanistic TMDD model and explore its mathematical structure in detail.

12 References

Gibiansky, L., & Gibiansky, E. (2009). Target-mediated drug disposition model: Approximations, identifiability of model parameters and applications to the population pharmacokinetic-pharmacodynamic modeling of biologics. Expert Opinion on Drug Metabolism & Toxicology, 5(7), 803–812. https://doi.org/10.1517/17425250903030043

Gibiansky, L., Gibiansky, E., Kakkar, T., & Ma, P. (2008). Approximations of the target-mediated drug disposition model and identifiability of model parameters. Journal of Pharmacokinetics and Pharmacodynamics, 35(5), 573–591. https://doi.org/10.1007/s10928-008-9102-8

Levy, G. (1994). Pharmacologic target-mediated drug disposition. Clinical Pharmacology & Therapeutics, 56(3), 248–252. https://doi.org/10.1038/clpt.1994.134

Peletier, L. A., & Gabrielsson, J. (2012). Dynamics of target-mediated drug disposition: Characteristic profiles and parameter identification. Journal of Pharmacokinetics and Pharmacodynamics, 39(5), 429–451. https://doi.org/10.1007/s10928-012-9260-6

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--- title: "TMDD Part 1: Understanding Target-Mediated Drug Disposition" execute: error: false cache: true engine: julia author: - Vijay Ivaturi - Srayant Gayam format: html: include-in-header: text: | <style> .header-container { width: 100%; } .logo { display: block; width: 150px; margin-left: -2.5%; } </style> <link rel = "shortcut icon" href = "https://pumas.ai/favicon.ico" /> include-before-body: text: | <div class="header-container"> <img src="https://pumas-assets.s3.amazonaws.com/CompanyLogos/PumasAI/RGB/PNG/Pumas+AI+Primary%404x.png" alt="Pumas Logo" class="logo"> </div> self-contained-math: true anchor-sections: true theme: default toc: true toc-depth: 4 toc-expand: 2 toc-location: left toc-title: Contents number-sections: true code-summary: Show/Hide Code code-tools: caption: Download tutorial source: true fig-format: svg fig-width: 8 fig-height: 6 license: CC BY-SA 4.0 bibliography: references.bib csl: https://raw.githubusercontent.com/citation-style-language/styles/master/apa.csl --- ## Introduction This is the first part of a comprehensive tutorial series on Target-Mediated Drug Disposition (TMDD) modeling in Pumas. In this part, we focus on building intuition for TMDD before diving into complex models. **Tutorial Series Overview:** 1. **Part 1: Understanding TMDD** (this tutorial) - Conceptual foundations and parameter meaning 1b. **Part 1b: TMDD Dynamics Through the Four Phases** - Deep dive into phase characterization 2. **Part 2: The Full TMDD Model** - Complete mechanistic model implementation 3. **Part 3: TMDD Approximations** - QE, QSS, and Michaelis-Menten models 4. **Part 4: Special Cases and Extensions** - Advanced TMDD scenarios 5. **Part 5: Practical Workflows** - Fitting, diagnostics, and model selection ## Learning Objectives By the end of this tutorial, you will be able to: - Explain what TMDD is and why it matters in drug development - Recognize TMDD signatures in pharmacokinetic data - Understand the biological basis of ligand-receptor binding kinetics - Interpret key TMDD parameters ($K_D$, $k_{on}$, $k_{off}$, $k_{syn}$, $k_{deg}$, $k_{int}$) - Distinguish between linear and nonlinear PK behavior ## Libraries ```{julia} using Pumas using PumasUtilities using AlgebraOfGraphics using CairoMakie using DataFrames using Random ``` ## What is Target-Mediated Drug Disposition? Target-Mediated Drug Disposition (TMDD) is a pharmacokinetic phenomenon where the binding of a drug (ligand) to its pharmacological target (receptor) significantly influences the drug's disposition in the body. This concept was formally introduced by @levy1994. The key insight is that for drugs with high affinity to their targets, the target itself becomes a meaningful elimination pathway. When target binding is saturable and the target concentration is pharmacologically relevant, the drug exhibits nonlinear pharmacokinetics. ### When Does TMDD Matter? TMDD is most commonly observed with: - **Monoclonal antibodies (mAbs)** binding to soluble or membrane-bound targets - **High-affinity small molecules** with potent target engagement - **Therapeutic proteins** interacting with cell surface receptors - **Antibody-drug conjugates (ADCs)** :::{.callout-note} ## The TMDD Signature A hallmark of TMDD is **dose-dependent clearance**: at low doses, clearance is high because target-mediated elimination dominates. At high doses (when targets are saturated), clearance decreases to reflect only non-specific elimination pathways. ::: ## The Biology of Ligand-Receptor Binding Before we can model TMDD, we need to understand the underlying biology. Consider a drug (the **ligand**, $L$) entering the body and encountering its pharmacological target (the **receptor**, $R$). ### The Binding Process The ligand-receptor interaction follows reversible binding kinetics: $$ L + R \underset{k_{off}}{\stackrel{k_{on}}{\rightleftharpoons}} LR $$ Where: - $L$ = Free ligand (drug) concentration - $R$ = Free receptor (target) concentration - $LR$ = Ligand-receptor complex concentration - $k_{on}$ = Association rate constant (units: concentration$^{-1}$ time$^{-1}$) - $k_{off}$ = Dissociation rate constant (units: time$^{-1}$) The equilibrium dissociation constant, $K_D$, characterizes the binding affinity: $$ K_D = \frac{k_{off}}{k_{on}} $$ A **lower** $K_D$ indicates **higher** binding affinity. Typical values for mAbs are in the nanomolar to picomolar range. ### Receptor Dynamics In the body, receptors are not static. They are continuously synthesized and degraded: $$ \emptyset \xrightarrow{k_{syn}} R \xrightarrow{k_{deg}} \emptyset $$ Where: - $k_{syn}$ = Zero-order synthesis rate of receptor (units: concentration/time) - $k_{deg}$ = First-order degradation rate constant of receptor (units: time$^{-1}$) At baseline (before drug administration), the receptor reaches a steady-state concentration: $$ R_0 = \frac{k_{syn}}{k_{deg}} $$ :::{.callout-tip} ## Parameter Interpretation The baseline receptor concentration $R_0$ is determined by the ratio of synthesis to degradation rates. This value is crucial for understanding the magnitude of TMDD effects. ::: ### Complex Fate: Internalization The ligand-receptor complex does not persist indefinitely. It is typically internalized (endocytosed) and degraded at a rate $k_{int}$: $$ LR \xrightarrow{k_{int}} \text{degradation products} $$ The relative magnitude of $k_{int}$ versus $k_{deg}$ determines whether target engagement accelerates or decelerates elimination: - If $k_{int} > k_{deg}$: The complex is cleared faster than free receptor, leading to **increased clearance** at low doses - If $k_{int} < k_{deg}$: The complex is more stable, potentially leading to accumulation ## Visualizing Linear vs. TMDD Kinetics Let's build intuition by comparing linear PK (no target binding) with TMDD kinetics. ### A Simple Linear Model First, we'll create a simple one-compartment model with linear elimination: ```{julia} linear_model = @model begin @param begin tvCl ∈ RealDomain(lower = 0) tvVc ∈ RealDomain(lower = 0) end @pre begin Cl = tvCl Vc = tvVc end @dynamics begin Central' = -Cl / Vc * Central end @derived begin Concentration = @. Central / Vc end end ``` ### A Simple TMDD Model Now, let's create a one-compartment TMDD model with target-mediated elimination: ```{julia} tmdd_simple_model = @model begin @param begin tvCl ∈ RealDomain(lower = 0) # Linear clearance tvVc ∈ RealDomain(lower = 0) # Central volume tvKsyn ∈ RealDomain(lower = 0) # Receptor synthesis rate tvKdeg ∈ RealDomain(lower = 0) # Receptor degradation rate tvKon ∈ RealDomain(lower = 0) # Association rate tvKoff ∈ RealDomain(lower = 0) # Dissociation rate tvKint ∈ RealDomain(lower = 0) # Internalization rate end @pre begin Cl = tvCl Vc = tvVc Ksyn = tvKsyn Kdeg = tvKdeg Kon = tvKon Koff = tvKoff Kint = tvKint end @dosecontrol begin bioav = (Ligand = 1 / tvVc,) # Dose enters as concentration end @init begin Receptor = Ksyn / Kdeg # Baseline receptor level (R0) end @vars begin R0 := Ksyn / Kdeg Ltot := Ligand + Complex Rtot := Receptor + Complex end @dynamics begin Ligand' = -(Cl / Vc) * Ligand - Kon * Ligand * Receptor + Koff * Complex Receptor' = Ksyn - Kdeg * Receptor - Kon * Ligand * Receptor + Koff * Complex Complex' = Kon * Ligand * Receptor - Koff * Complex - Kint * Complex end @derived begin Free_Ligand = @. Ligand Free_Receptor = @. Receptor Complex_Conc = @. Complex end end ``` ### Comparing the Models Let's simulate both models across a range of doses to see how TMDD manifests as nonlinear kinetics: ```{julia} #| code-fold: true # Parameters for linear model param_linear = (tvCl = 0.01, tvVc = 3.0) # Parameters for TMDD model based on Gibiansky et al. # These values represent typical mAb parameters param_tmdd = ( tvCl = 0.006, # Linear clearance: ~0.15 L/day tvVc = 3.0, # Central volume: 3 L (plasma volume) tvKsyn = 0.04, # Synthesis rate (nM/hr) → ~1 nmol/day tvKdeg = 0.008, # Degradation rate (hr⁻¹) → R0 = 5 nM, t½ ~87 hr tvKon = 0.33, # Association rate (nM⁻¹ hr⁻¹) → ~8 (nM)⁻¹/day tvKoff = 0.33, # Dissociation rate (hr⁻¹) → KD = 1 nM tvKint = 0.002, # Internalization rate (hr⁻¹) → ~0.04/day ) # Create populations with different doses # Wide dose range to show saturation of target-mediated elimination doses = [0.5, 5, 50, 500] # nmol (spanning 3 orders of magnitude) Random.seed!(42) pop_linear = [ Subject( id = i, events = DosageRegimen(dose, time = 0), observations = (Concentration = nothing,), ) for (i, dose) in enumerate(doses) ] pop_tmdd = [ Subject( id = i, events = DosageRegimen(dose, time = 0, cmt = 1), observations = ( Free_Ligand = nothing, Free_Receptor = nothing, Complex_Conc = nothing, ), ) for (i, dose) in enumerate(doses) ] # Simulate (extended time for slower mAb kinetics) obstimes = 0.1:2:672 # Up to 4 weeks sim_linear = simobs(linear_model, pop_linear, param_linear, obstimes = obstimes) sim_tmdd = simobs(tmdd_simple_model, pop_tmdd, param_tmdd, obstimes = obstimes) ``` Now let's visualize the difference: ```{julia} #| code-fold: true #| fig-cap: "Comparison of linear PK (left) versus TMDD kinetics (right). Linear PK shows superimposable dose-normalized curves. TMDD shows dose-dependent behavior with faster decline at lower doses." # Convert simulation results to DataFrames df_linear = DataFrame(sim_linear) df_tmdd = DataFrame(sim_tmdd) # Create dose label mapping dose_labels = DataFrame(id = string.(1:4), Dose = string.(doses) .* " nmol") # Add dose labels and calculate dose-normalized concentrations df_linear = innerjoin(df_linear, dose_labels, on = :id) df_linear.Conc_Norm = df_linear.Concentration ./ parse.(Float64, replace.(df_linear.Dose, " nmol" => "")) df_linear.Model .= "Linear PK" df_tmdd = innerjoin(df_tmdd, dose_labels, on = :id) df_tmdd.Conc_Norm = df_tmdd.Free_Ligand ./ parse.(Float64, replace.(df_tmdd.Dose, " nmol" => "")) df_tmdd.Model .= "TMDD" # Combine for faceted plot df_combined = vcat( select(df_linear, :time, :Dose, :Conc_Norm, :Model), select(df_tmdd, :time, :Dose, :Conc_Norm, :Model), ) plt = data(df_combined) * mapping( :time => "Time (hr)", :Conc_Norm => "Dose-Normalized Concentration"; color = :Dose, layout = :Model, ) * visual(Lines; linewidth = 2) draw(plt; axis = (yscale = Makie.pseudolog10,)) ``` :::{.callout-important} ## Key Observation In linear PK (left panel), dose-normalized concentrations **superimpose** regardless of dose level. In TMDD (right panel), dose-normalized curves **diverge**: lower doses show faster terminal decline because target-mediated clearance dominates. ::: ## Recognizing TMDD in Your Data When analyzing PK data, several signatures suggest TMDD may be present: ### Dose-Dependent Half-Life The apparent terminal half-life changes with dose: - **At low doses**: Short half-life (target-mediated elimination dominates) - **At high doses**: Long half-life (linear elimination dominates after target saturation) ```{julia} #| code-fold: true #| fig-cap: "Concentration-time profiles showing dose-dependent terminal slope characteristic of TMDD." plt = data(df_tmdd) * mapping( :time => "Time (hr)", :Free_Ligand => "Free Ligand Concentration"; color = :Dose => "Dose", ) * visual(Lines; linewidth = 2) draw( plt; axis = (yscale = Makie.pseudolog10, title = "TMDD: Dose-Dependent Terminal Kinetics"), ) ``` ### Non-Parallel Decline On a semi-log plot, concentration-time curves at different doses are **not parallel** during the terminal phase. This contrasts with linear kinetics where all doses show parallel decline. ### Rapid Initial Decline Followed by Slower Terminal Phase TMDD often produces a characteristic **multiphasic** profile. @peletier2012 identified four distinct phases (A, B, C, D) in TMDD concentration-time curves: 1. **Phase A**: Rapid initial binding (target association) 2. **Phase B**: Linear decline with saturated target 3. **Phase C**: Non-linear transition as target recovers 4. **Phase D**: Terminal linear phase (often internalization-limited) This four-phase characterization is explored in detail in **Part 1b: TMDD Dynamics Through the Four Phases**, which provides comprehensive visualizations and parameter identification guidance. ### Disproportionate Increase in AUC with Dose For a linear drug, AUC increases proportionally with dose. With TMDD, AUC increases **more than proportionally** at higher doses because target-mediated clearance becomes saturated. ## Understanding TMDD Parameters Let's explore how each parameter affects the PK profile through interactive simulations. ### Effect of Binding Affinity ($K_D$) The dissociation constant $K_D = k_{off}/k_{on}$ determines binding affinity. Lower $K_D$ means tighter binding. ```{julia} #| code-fold: true #| fig-cap: "Effect of binding affinity (KD) on TMDD profiles. Tighter binding (lower KD) leads to more pronounced target-mediated elimination." # Single dose, varying KD single_pop = [ Subject( id = 1, events = DosageRegimen(50, time = 0, cmt = 1), observations = ( Free_Ligand = nothing, Free_Receptor = nothing, Complex_Conc = nothing, ), ), ] kd_values = [0.1, 1.0, 5.0, 20.0] # nM; Lower = higher affinity # Build DataFrame with all KD variations df_kd = DataFrame() for kd in kd_values param_kd = ( tvCl = 0.006, tvVc = 3.0, tvKsyn = 0.04, tvKdeg = 0.008, tvKon = 0.33, tvKoff = kd * 0.33, # Koff = KD * Kon tvKint = 0.002, ) sim_kd = simobs(tmdd_simple_model, single_pop, param_kd, obstimes = obstimes) sim_df = DataFrame(sim_kd) sim_df.KD .= "KD = $kd nM" append!(df_kd, sim_df) end plt = data(df_kd) * mapping( :time => "Time (hr)", :Free_Ligand => "Free Ligand Concentration"; color = :KD => "KD (nM)", ) * visual(Lines; linewidth = 2) draw(plt; axis = (yscale = Makie.pseudolog10, title = "Effect of Binding Affinity (KD)")) ``` ### Effect of Receptor Concentration ($R_0$) The baseline receptor level $R_0 = k_{syn}/k_{deg}$ determines the magnitude of TMDD. Higher receptor concentrations mean more drug can be eliminated via target binding. ```{julia} #| code-fold: true #| fig-cap: "Effect of baseline receptor concentration (R0) on TMDD profiles. Higher R0 leads to greater target-mediated elimination." r0_values = [1, 5, 15, 30] # Different R0 levels (nM) # Build DataFrame with all R0 variations df_r0 = DataFrame() for r0 in r0_values param_r0 = ( tvCl = 0.006, tvVc = 3.0, tvKsyn = r0 * 0.008, # Ksyn = R0 * Kdeg tvKdeg = 0.008, tvKon = 0.33, tvKoff = 0.33, tvKint = 0.002, ) sim_r0 = simobs(tmdd_simple_model, single_pop, param_r0, obstimes = obstimes) sim_df = DataFrame(sim_r0) sim_df.R0 .= "R0 = $r0 nM" append!(df_r0, sim_df) end plt = data(df_r0) * mapping( :time => "Time (hr)", :Free_Ligand => "Free Ligand Concentration"; color = :R0 => "R0 (nM)", ) * visual(Lines; linewidth = 2) draw( plt; axis = (yscale = Makie.pseudolog10, title = "Effect of Baseline Receptor Level (R0)"), ) ``` ### Effect of Internalization Rate ($k_{int}$) The internalization rate determines how quickly the ligand-receptor complex is eliminated. ```{julia} #| code-fold: true #| fig-cap: "Effect of internalization rate on TMDD profiles. Faster internalization increases target-mediated clearance." kint_values = [0.001, 0.002, 0.008, 0.02] # hr⁻¹ # Build DataFrame with all kint variations df_kint = DataFrame() for kint in kint_values param_kint = ( tvCl = 0.006, tvVc = 3.0, tvKsyn = 0.04, tvKdeg = 0.008, tvKon = 0.33, tvKoff = 0.33, tvKint = kint, ) sim_kint = simobs(tmdd_simple_model, single_pop, param_kint, obstimes = obstimes) sim_df = DataFrame(sim_kint) sim_df.kint .= "kint = $kint hr⁻¹" append!(df_kint, sim_df) end plt = data(df_kint) * mapping( :time => "Time (hr)", :Free_Ligand => "Free Ligand Concentration"; color = :kint => "kint (hr⁻¹)", ) * visual(Lines; linewidth = 2) draw( plt; axis = (yscale = Makie.pseudolog10, title = "Effect of Internalization Rate (kint)"), ) ``` ## Parameter Summary Table The following table summarizes the key TMDD parameters and their biological interpretation: | Parameter | Symbol | Units | Biological Meaning | |:--------------------- |:---------:|:------------------:|:----------------------------------------------------- | | Association rate | $k_{on}$ | nM$^{-1}$hr$^{-1}$ | Rate of ligand-receptor binding | | Dissociation rate | $k_{off}$ | hr$^{-1}$ | Rate of complex dissociation | | Dissociation constant | $K_D$ | nM | Binding affinity ($K_D = k_{off}/k_{on}$) | | Synthesis rate | $k_{syn}$ | nM/hr | Rate of receptor production | | Degradation rate | $k_{deg}$ | hr$^{-1}$ | Rate of free receptor turnover | | Internalization rate | $k_{int}$ | hr$^{-1}$ | Rate of complex elimination | | Baseline receptor | $R_0$ | nM | Steady-state receptor level ($R_0 = k_{syn}/k_{deg}$) | :::{.callout-tip} ## Typical Parameter Ranges for Monoclonal Antibodies Based on @gibiansky2008 and @gibiansky2009, along with clinical experience with therapeutic mAbs: | Parameter | Typical Range | Units | Notes | |:--------- |:------------- |:------------------- |:------------------------------------- | | $CL$ | 0.1 - 0.5 | L/day | Non-specific linear clearance | | $V_c$ | 2 - 5 | L | Approximately plasma volume | | $K_D$ | 0.01 - 10 | nM | Very high affinity | | $K_{SS}$ | 0.1 - 10 | nM | $K_{SS} = (k_{off} + k_{int})/k_{on}$ | | $k_{on}$ | 0.1 - 10 | nM$^{-1}$day$^{-1}$ | Association rate | | $k_{off}$ | 0.01 - 10 | day$^{-1}$ | Dissociation rate | | $R_0$ | 0.1 - 100 | nM | Target-dependent | | $k_{deg}$ | 0.1 - 1 | day$^{-1}$ | Receptor turnover | | $k_{int}$ | 0.01 - 1 | day$^{-1}$ | Often similar to $k_{deg}$ | The ratio of $k_{int}/k_{deg}$ determines whether binding increases or decreases target turnover. ::: ## When to Suspect TMDD Consider TMDD modeling when you observe: 1. **Dose-dependent clearance** decreasing with increasing dose 2. **Non-superimposable dose-normalized profiles** 3. **Rapid initial decline** at low doses followed by slower terminal phase 4. **Greater than dose-proportional AUC increases** 5. **Biological plausibility**: high-affinity target with pharmacologically relevant concentrations :::{.callout-warning} ## Not All Nonlinearity is TMDD Other sources of nonlinear PK include: - Saturable absorption (flip-flop kinetics) - Saturable protein binding - Capacity-limited hepatic metabolism - Time-dependent autoinduction or inhibition - Formation of anti-drug antibodies Careful mechanistic analysis is needed to distinguish these phenomena. ::: ## Summary In this tutorial, we covered the conceptual foundations of TMDD: - **TMDD occurs** when high-affinity target binding significantly affects drug disposition - **Key components** are the ligand, receptor, and complex, each with associated rate constants - **The signature of TMDD** is dose-dependent clearance and non-superimposable profiles - **Parameters have biological meaning**: $K_D$ reflects affinity, $R_0$ reflects target abundance, $k_{int}$ reflects complex elimination In Part 2, we will implement the full mechanistic TMDD model and explore its mathematical structure in detail. ## References

TMDD Part 1: Understanding Target-Mediated Drug Disposition

1 Introduction

2 Learning Objectives

3 Libraries

4 What is Target-Mediated Drug Disposition?

4.1 When Does TMDD Matter?

5 The Biology of Ligand-Receptor Binding

5.1 The Binding Process

5.2 Receptor Dynamics

5.3 Complex Fate: Internalization

6 Visualizing Linear vs. TMDD Kinetics

6.1 A Simple Linear Model

6.2 A Simple TMDD Model

6.3 Comparing the Models

7 Recognizing TMDD in Your Data

7.1 Dose-Dependent Half-Life

7.2 Non-Parallel Decline

7.3 Rapid Initial Decline Followed by Slower Terminal Phase

7.4 Disproportionate Increase in AUC with Dose

8 Understanding TMDD Parameters

8.1 Effect of Binding Affinity (\(K_D\))

8.2 Effect of Receptor Concentration (\(R_0\))

8.3 Effect of Internalization Rate (\(k_{int}\))

9 Parameter Summary Table

10 When to Suspect TMDD

11 Summary

12 References

Reuse

Parameter	Symbol	Units	Biological Meaning
Association rate	\(k_{on}\)	nM\(^{-1}\)hr\(^{-1}\)	Rate of ligand-receptor binding
Dissociation rate	\(k_{off}\)	hr\(^{-1}\)	Rate of complex dissociation
Dissociation constant	\(K_D\)	nM	Binding affinity (\(K_D = k_{off}/k_{on}\))
Synthesis rate	\(k_{syn}\)	nM/hr	Rate of receptor production
Degradation rate	\(k_{deg}\)	hr\(^{-1}\)	Rate of free receptor turnover
Internalization rate	\(k_{int}\)	hr\(^{-1}\)	Rate of complex elimination
Baseline receptor	\(R_0\)	nM	Steady-state receptor level (\(R_0 = k_{syn}/k_{deg}\))

Parameter	Typical Range	Units	Notes
\(CL\)	0.1 - 0.5	L/day	Non-specific linear clearance
\(V_c\)	2 - 5	L	Approximately plasma volume
\(K_D\)	0.01 - 10	nM	Very high affinity
\(K_{SS}\)	0.1 - 10	nM	\(K_{SS} = (k_{off} + k_{int})/k_{on}\)
\(k_{on}\)	0.1 - 10	nM\(^{-1}\)day\(^{-1}\)	Association rate
\(k_{off}\)	0.01 - 10	day\(^{-1}\)	Dissociation rate
\(R_0\)	0.1 - 100	nM	Target-dependent
\(k_{deg}\)	0.1 - 1	day\(^{-1}\)	Receptor turnover
\(k_{int}\)	0.01 - 1	day\(^{-1}\)	Often similar to \(k_{deg}\)